Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees

TL;DR

This paper compares the enumeration of different classes of rooted binary phylogenetic networks, revealing asymptotic counts and relationships among general galled trees, time-consistent galled trees, and simplex time-consistent galled trees.

Contribution

It provides new enumerative results and asymptotic counts for two additional classes of galled trees, expanding understanding of their combinatorial properties.

Findings

Asymptotic count of general and time-consistent galled trees are identical for fixed galls.

Simplex time-consistent galled trees are less numerous than other classes asymptotically.

Enumerative formulas and recursions for small networks are also provided.

Abstract

Rooted binary phylogenetic networks are extensions of rooted binary trees, adding reticulation nodes that are designed to represent evolutionary processes that involve hybridization events. Enumerative combinatorics studies have counted leaf-labeled phylogenetic networks in a variety of classes, finding that when the number of reticulations is fixed, the time-consistent galled trees are asymptotically less numerous than each of several network classes that had been previously examined. Here we provide enumerative results on two additional network classes: general galled trees and simplex time-consistent galled trees. We show that for a fixed number of galls, as the number of leaves goes to infinity, the asymptotic count of general galled trees is identical to that of time-consistent galled trees, whereas the count of simplex time-consistent galled trees is smaller. If the number of galls is not restricted, then the asymptotic approximations all differ: simplex time-consistent galled trees are less numerous than time-consistent galled trees, which are in turn less numerous than general galled trees. We also report a variety of additional results: recursions to count the studied networks with small numbers of leaves a fixed number of galls, as well as enumerative results for unlabeled networks in the classes that we investigate.